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Unformatted text preview: 16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #8 - Principle of Virtual Displacements Raul Radovitzky March 3, 2005 Principle of Virtual Displacements Consider a body in equilibrium. We know that the stress field must satisfy the differential equations of equilibrium. Multiply the differential equations of equilibrium by an arbitrary displacement field u i : ji,j + f i u i = 0 (1) Note that the field u i is NOT the actual displacement field u i corresponding to the solution of the problem but a virtual displacement field. Therefore, equation ( 1 ) can be interpreted as the local expression of virtual work done by the actual stresses and the body forces on the virtual displacement u i and that it must be zero. The total virtual work done on the body is obtained by integration over the volume: ji,j + f i u i dV = 0 (2) V 1 and it must also be zero since the integrand is zero everywhere in the domain. ji,j u i dV + f i u i dV = 0 (3) V V ji u i ,j ji u i,j dV + f i u i dV = 0 (4) V V ji u i n j dS ij ij dV + f i u i dV = 0 (5) S V V The integral over the surface can be decomposed into two: an integral over the portion of the boundary where the actual external surface loads (tractions) are specified S t and an integral over the portion of the boundary where the displacements are specified (supports) S u . This assumes that these sets are disjoint and complementary, i.e., S = S u S t , S u S t = (6) S t t i u i dS + S u ji u i n j dS V ij ij dV + V f i u i dV = 0 (7) We will require that the virtual displacements u i vanish on S u , i.e., that the virtual displacement field satisfy the homogeneous essential boundary condi- tions : u i ( x j ) = 0 , if x j S u (8) Then, the second integral vanishes. The resulting expression is a statement of the Principle of Virtual Displacements...
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